1210.4088 (Denis Borisov et al.)

On the spectrum of deformations of compact double-sided flat
hypersurfaces    [PDF]

Denis Borisov, Pedro Freitas

We study the asymptotic behaviour of the eigenvalues of the Laplace-Beltrami operator on a compact hypersurface in \mathds{R}^{n+1} as it is flattened into a singular double-sided flat hypersurface. We show that the limit spectral problem corresponds to the Dirichlet and Neumann problems on one side of this flat (Euclidean) limit, and derive an explicit three-term asymptotic expansion for the eigenvalues where the remaining two terms are of orders \varepsilon^2\log\varepsilon and \varepsilon^2.

View original: http://arxiv.org/abs/1210.4088